But denying things that have been said is different to fact checking vague statistics for eg as the voters themselves can clearly see or can clearly be shown that things have been lied about. If Trump says 20% of immigrants commit a crime within their first 5 years his diehards will take it as gospel or at the very least ignore anything that suggests different but seeing a video of Trump saying something in an interview and 2 seconds later seeing a clip of Pence denying it was ever said will get through that denial and register even if they pretend otherwise.
I think the problem is many times issues aren't as black or white as the example you gave, so unless the fact checkers line up with one's ideology they will ignore it and feel like they have an agenda.
Many so called lies are what most fact checkers would call half truths. And example of this is saying that Chicago has the most murders is a half truth. While it's true Chicago has the most murders in terms of sheer numbers, when you factor Chicago has 2.7M people and we generally comparing them to cities with like 1/4 and 1/5 of that population it's sort of saying something is true but not putting it in proper context. I am guessing many who want to view Chicago as the worst city possible feel this is the fact checkers having an agenda.